A study in a particular state found that only 5% of the voters have Doctoral degrees. In a random sample of 200 voters, let x be the number who have doctoral degrees.
6) Find the mean of x.
7) Find the standard deviation of x.
8) Find the z-score for x > 10.5
9) What is the probability that more than 10.5 voters in a sample of 200 voters will hold a Doctoral degree?
10) What is the probability that at least 10.5 voters in a sample of 200 voters will hold a Doctoral degree?
To answer these questions, we need to use the binomial distribution formula since we are dealing with a binary outcome (having a doctoral degree or not having a doctoral degree) and want to calculate probabilities.
First, let's define the parameters for the binomial distribution formula:
- n is the number of trials (sample size) = 200 (given)
- p is the probability of success in each trial (proportion of voters with a doctoral degree) = 0.05 (given, 5%)
- x is the number of successes (number of voters with a doctoral degree)
6) To find the mean of x, we use the formula: μ = n * p
Substituting the given values: μ = 200 * 0.05 = 10
Therefore, the mean of x is 10.
7) To find the standard deviation of x, we use the formula: σ = sqrt(n * p * (1 - p))
Substituting the given values: σ = sqrt(200 * 0.05 * 0.95) = sqrt(9.5) ≈ 3.08
Therefore, the standard deviation of x is approximately 3.08.
8) To find the z-score for x > 10.5, we use the formula: z = (x - μ) / σ
Substituting the given values: z = (10.5 - 10) / 3.08 ≈ 0.163
Therefore, the z-score for x > 10.5 is approximately 0.163.
9) The probability that more than 10.5 voters in a sample of 200 voters will hold a doctoral degree can be calculated using the cumulative probability function of the binomial distribution. Since probabilities are generally calculated for discrete values, we need to use the "greater than" inequality as follows:
P(x > 10.5) = 1 - P(x ≤ 10) = 1 - Σ(k=0 to 10) C(n,k) * p^k * (1-p)^(n-k)
Here, C(n,k) represents the combination function.
10) The probability that at least 10.5 voters in a sample of 200 voters will hold a doctoral degree can be calculated similarly to question 9 by using the cumulative probability function:
P(x ≥ 10.5) = 1 - P(x < 10.5) = 1 - Σ(k=0 to 10) C(n,k) * p^k * (1-p)^(n-k)
To obtain precise numerical values for these probabilities, you can use statistical software or a binomial distribution calculator, which can perform the necessary calculations using the formulas provided above.